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Specific wave profiles and closed-form soliton solutions for generalized nonlinear wave equation in(3+1)-dimensions with gas bubbles in hydrodynamics and fluids

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Journal of Ocean Engineering and Science 2023年第1期8卷 91-102页

作者： Sachin Kumar ihsanullah hamid M.A.Abdou Department of Mathematics Faculty of Mathematical SciencesUniversity of DelhiDelhi 110007India Department of Mathematics College of SciencesUniversity of BishaP.O.Box 344Bisha 61922Saudi Arabia Physics Department Faculty of ScienceMansoura UniversityMansoura 35516Egypt

Nonlinear evolution equations(NLEEs)are frequently employed to determine the fundamental principles of natural *** equations are studied extensively in nonlinear sciences,ocean physics,fluid dynamics,plasma physics,scientific applications,and marine *** generalized exponen-tial rational function(GERF)technique is used in this article to seek several closed-form wave solutions and the evolving dynamics of different wave profiles to the generalized nonlinear wave equation in(3+1)dimensions,which explains several more nonlinear phenomena in liquids,including gas bubbles.A large number of closed-form wave solutions are generated,including trigonometric function solutions,hyper-bolic trigonometric function solutions,and exponential rational functional *** the dynamics of distinct solitary waves,a variety of soliton solutions are obtained,including single soliton,multi-wave structure soliton,kink-type soliton,combo singular soliton,and singularity-form wave *** de-termined solutions have never previously been *** dynamical wave structures of some analyt-ical solutions are graphically demonstrated using three-dimensional graphics by providing suitable values to free *** technique can also be used to obtain the soliton solutions of other well-known equations in engineering physics,fluid dynamics,and other fields of nonlinear sciences.

关键词： Closed-form solutions Dynamical wave patterns Analytic solutions Nonlinear wave equation GERF Method Solitary waves Solitons

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Balancing Chemical Equations by Systems of Linear Equations

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Applied Mathematics 2019年第7期10卷 521-526页

作者： ihsanullah hamid Department of Mathematics University of Nangarhar Jalalabad Nangarhar Afghanistan

In this paper, a formal and systematic method for balancing chemical reaction equations was presented. The results satisfy the law of conservation of matter, and confirm that there is no contradiction to the existing way(s) of balancing chemical equations. A chemical reaction which possesses atoms with fractional oxidation numbers that have unique coefficients was studied. In this paper, the chemical equations were balanced by representing the chemical equation into systems of linear equations. Particularly, the Gauss elimination method was used to solve the mathematical problem with this method, it was possible to handle any chemical reaction with given reactants and products.

关键词： Chemical Reaction Linear Equations Balancing Chemical Equations Matrix Gauss Elimination Method

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